Abstract: for (transient) one dimensional random walk in random environment, conditions are known that ensure an annealed CLT. One then also have a quenched CLT, with a different (environment dependent) centering.
In higher dimensions, annealed CLT's have been derived in the ballistic case by Sznitman. We prove that in dimension 4 or more, annealed CLT's together with a mild integrability condition imply a quenched CLT. The proof is based on controlling the intersections of two RWRE paths in the same environment.
We will discuss a strong law of large numbers, an annealed CLT, and
the limit law of the ``environment viewed from the particle" for transient
random walks on a strip (product of Z with a finite set). The model was
introduced by Bolthausen and Goldsheid and includes in particular RWRE
with bounded jumps on the line as well as some one-dimensional RWRE with a
memory.
We consider a nonlinear Schr\"odinger equation (NLS) with random
coefficients, in a regime of separation of scales corresponding to
diffusion approximation. The primary goal is to propose and
study an efficient numerical scheme in this framework. We use a
pseudo-spectral splitting scheme and we establish the order of the
global error. In particular we show that we can take an integration step
larger than the smallest scale of the problem, here the correlation
length of the random medium. We study
the asymptotic behavior of the numerical solution in the diffusion
approximation regime.
Let G be a transient graph, and flip a fair coin at each vertex.
This gives a distribution P. Now start a random walk from a vertex v, and
retoss the coin at each visited vertex, this time with probability 0.75
for heads and probability 0.25 for tails. The eventual configuration of
the coins gives a distribution Q. Are P and Q absolutely continuous w.r.t.
each other? are they singular? (i.e. can you tell whether a random walker
had tampered with the coins or not?) In the talk I'll answer to this
question for various graphs and various types of random walk. Based on
joint work with Y. Peres.
Mean-field theory is one of the most standard tools used by
physicists to analyze phase transitions in realistic systems. However,
regarding rigorous proofs, the link to mean-field theory has been
limited to asymptotic statements which do not yield enough control
of the actual systems. In this talk I will describe a new approach to
this set of problems -- developed jointly with Lincoln Chayes and
Nicolas Crawford -- that overcomes this hurdle in a rather elegant
way. As a conclusion, I will show that a general, ferromagnetic
nearest neighbor spin system on Z^d undergoes a first order phase
transition whenever the mean-field theory indicates one, provided
the dimension d is sufficiently large. Extensions to systems with non
nearest neighbor interactions will also be discussed.
Given the measure on random walk paths $P_0$ and a Hamiltonian $H$ the Gibbs perturbation of $H$ defined by
$$\frac{dP_{\beta,t}}{dP_0}=Z^{-1}_{\beta,t}\exp\{\beta H(x)\}$$
with
$$Z_{\beta,t}=\int \exp\{-\beta H(x)\}dP_0(x)$$
gives a new measure on paths $x$ which can be viewed as polymers.
In the case $H(x)=\int_0^t\delta_0(x_{s})ds(\int_0^t\delta_0(x_{s})dW_s)$ we say the resulting measure is concentrated on "homopolymers" ("heteropolymers") and are interested in the influence of dimension and $\beta$ on their behavior.